arXiv:1208.1054v4 [q-bio.MN] 24 Feb 2013
Transfer Functions for Protein Signal Transduction:
Application to a Model of Striatal Neural Plasticity
Gabriele Scheler
Carl Correns Foundation for Mathematical Biology
Mountain View, Ca, USA
Abstract
We present a novel formulation for biochemical reaction networks in the context
of protein signal transduction. The model consists of input-output transfer functions,
which are derived from differential equations, using stable equilibria. We select a set
of ’source’ species, which are interpreted as input signals. Signals are transmitted to
all other species in the system (the ’target’ species) with a specific delay and with a
specific transmission strength. The delay is computed as the maximal reaction time
until a stable equilibrium for the target species is reached, in the context of all other
reactions in the system. The transmission strength is the concentration change of the
target species. The computed input-output transfer functions can be stored in a
matrix, fitted with parameters, and even recalled to build dynamical models on the
basis of state changes. By separating the temporal and the magnitudinal domain we
can greatly simplify the computational model, circumventing typical problems of
complex dynamical systems. The transfer function transformation of biochemical
reaction systems can be applied to mass-action kinetic models of signal transduction.
The paper shows that this approach yields significant novel insights while remaining a
fully testable and executable dynamical model for signal transduction. In particular
we can deconstruct the complex system into local transfer functions between
individual species. As an example, we examine modularity and signal integration
using a published model of striatal neural plasticity. The modularizations that
emerge correspond to a known biological distinction between calcium-dependent and
cAMP-dependent pathways. Remarkably, we found that overall interconnectedness
depends on the magnitude of inputs, with higher connectivity at low input
concentrations and significant modularization at moderate to high input
concentrations. This general result, which directly follows from the properties of
individual transfer functions, contradicts notions of ubiquitous complexity by showing
input-dependent signal transmission inactivation.
Introduction
Biochemical reaction systems are usually conceptualized as dynamical systems - systems
that evolve in continuous time and may or may not receive additional input to the system.
Mathematically, this can be expressed by sets of ordinary differential equations (ODE), such
that rates of concentration changes correspond to mass-action kinetic parameters [1, 2].
In this paper we use existing mass-action dynamical systems to propose an alternate or
1
2
additional framework for modeling and interpretation of biochemical reaction systems. We
provide an algebraic analysis of biochemical reaction systems as a matrix of concentrations for
all species, given certain input concentrations. These concentrations correspond to steadystate amounts which are reached after a delay time, and the delay times can be measured
by the system as well.
We use an arbitrary published model [3] as an example for a ODE dynamical model of
biochemical reactions. The model simulates intracellular signal transduction from receptor
binding to molecular targets in different cellular compartments, as an important component
in the long-term regulation of protein expression implied in neural and synaptic plasticity.
In striatal neurons, both a calcium-dependent pathway and a cAMP-dependent pathway are
activated during the initiation of neural plasticity by NMDA/AMPA receptors and neuromodulator receptors such as dopamine D1 receptors [4, 5, 6]. Their effects and the integration
of signaling on common targets such as kinases and phosphatases have been the subject of
a number of computational models [7, 8, 9, 10, 11]. In particular, the role of the DARPP32 protein in striatal neurons in determining the outcome of membrane signaling has been
modeled by different groups, based on a common set of experimental data [12, 13, 3], cf.
[14, 15, 16, 17, 18]. Many similar models [19] have been developed in the last 10-12 years in
different areas of biology. Models with dozens or more of species have up to a 100 or more
equations and are consequently complex and difficult to understand as continuous dynamical
systems [20]. A transformation into a matrix-based formulation of input-output functions,
even at the cost of a loss of fast dynamical modeling, promises considerable gain of insight
and access to a different set of mathematical tools. Simple mass-action kinetic models may
be criticized for disregarding the real complexity of spatio-temporal molecular interactions.
Some alternatives use spatial grids and stochastic versions of biochemical reactions to capture this complexity [21, 22]. However, certain variations, such as compartmental modeling
with diffusion, altered kinetics for anchored proteins, or employing molecular kinetics as the
basis for binding constants may be employed within the mass-action kinetic framework to
achieve better correspondence with the biological reality. These variations can be directly
transferred to the proposed model as well.
In our approach, we identify input nodes, and then pre-compute the outcomes for all internal species (target species) in response to biological meaningful ranges and combinations
of inputs. This allows to analyze a biochemical reaction system under all possible input
conditions. The analysis can be done for arbitrary ODE models [19], provided minimal requirements on conservation properties are realized (cf. section “Methods”, [23]). The results
are stored as vectors or matrices (’systemic protein signaling functions’ (psfs)) and can be
fitted with functional parameters. It is an important aspect of the model that computations are done systemically. In section “Elementary Biochemical Reactions”, source-target
interactions are first analyzed in isolation (’elementary psfs’). They all constitute hyperbolic
saturation functions, therefore rate parameters can be uniformly translated into functional
parameters for signal transmission strength. But in a systemic context, source-target interactions change because of additional influences on the species from other equations in the
system (section “Systemic PSF Analysis”). Therefore a fitted systemic psf from A to B is
different from the elementary psf. The pre-computed, systemic psfs may be used to create state-change simulation models, i.e. discrete-time models, which can be compared with
continuous ODE models (section “Systemic Delay and State-change Dynamics”). What is
3
significant and novel about our analysis is that we can extract systemic transfer functions
from the complex system, and thereby dissect the system into parts. We can analyze the
transmission properties of individual species, compare their minimum and maximum values,
and the functional shape of their transmission strength. Specifically, we can show under
which circumstances a link is functional, i.e. actually transmits information (section “Computing Input-dependent Modularity”).
The analysis has a number of restrictions. An important restriction is that our model
does not allow for analysis of fast interactions below the resolution of settling into steadystate. The requirement of conservation of mass guarantees that for each input concentrations
will eventually settle to some equilibrium value, but due to the prevalence of feedback interactions, they may still produce transients or dampened oscillations. This means that
fast fluctuations of input will not be adequately simulated using pre-computed psf functions
alone. It is then necessary to refer to the underlying ODE model. The model is most suitable
for studying disease states, pharmacological interventions, genetic manipulations, miRNA interference, or any system conditions which fundamentally alter the presence or concentration
of molecular species. These conditions may then be tested either in steady-state or with a
sequence of sufficiently slow input constellations. A second restriction is that the model inherits parameter uncertainty from mass-action kinetic models. These parameters are derived
from experimental measurements, but typically with a high degree of uncertainty [24]. Our
analysis offers a clear distinction between elementary and systemic functional parameters
and explains why experimental measurements are so highly dependent on the systemic context. In this paper, we have treated elementary parameters as given by the underlying model
[3]. In principle, systemic functional interactions can be measured experimentally, and in
the model each interaction can be adjusted separately. This may offer a novel theoretical
approach towards finding adequate elementary rate parameters.
Materials and Methods
System Definition
A biochemical reaction system formulation for signal transduction contains two different
types of reactions:
1. complex formation
[A] + [B] ↔ [AB]
2. enzymatic reactions
[A] + [E] ↔ [AE] → [A∗] + E.
The system has concentrations for species A, B and E (A*, AE and AB can be calculated),
a set of kinetic rate parameters kon and koff for the forward and backward binding reactions
in complex formation, and kcat for the rate of enzymatic production. The equational structure, the kinetic parameters and the initial concentrations of the model [3] are reproduced in
Tables 1, 2, 3, 4, with slight modifications: Equation 40 was added to ’close the loop’ from
AMP to ATP and thus provide for conservation of all molecules in the system. Conservation
4
of mass is necessary in order for all species to reach equilibrium. It means that for any forward reaction there needs to be a reverse reaction, such that any species receives both input
and output (’weakly reversible system’, cf. [23]). This implies that pure loss reactions, like
endocytosis or diffusion across the cell membrane (secretion) cannot adequately be modeled
with this system, unless balancing reactions are added which make up for the loss. E.g.,
for endocytosis of a ligand-bound receptor, both the ligand and the receptor, possibly independently, have to be recycled, which means that bridging equations for receptor-ligand
dissolution in the endocytosed state, and input rates for receptors and ligands have to be
added. Secondly, the species PP1 and its interactions (4 equations) were left out, since they
contain a complex which dissolves into its 3 components in one step: this would require a,
fairly trivial, addition to the current psf system implementation. The system can be depicted
as a bipartite graph with nodes for species and nodes for reactions.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Da + D1R ↔ DaD1R
DaD1R + Gabc ↔ DaD1RGabc
Gabc + D1R ↔ GabcD1R
GabcD1R + Da ↔ DaD1RGabc
DaD1RGabc → DaD1R + GoaGT P + Gbc
GoaGT P → GoaGDP
GoaGDP + Gbc → Gabc
GoaGT P + AC5 ↔ AC5GoaGT P
AT P + AC5GoaGT P ↔ AC5GoaGT P AT P →
cAM P + AC5GoaGT P
AC5 + Ca ↔ AC5Ca
AC5Ca + GoaGT P ↔ AC5CaGoaGT P
AT P + AC5CaGoaGT P ↔ AC5CaGoaGT P AT P →
cAM P + AC5CaGoaGT P
P DE1 + Ca4CaM ↔ P DE1CaM
cAM P + P DE1CaM ↔ P DE1CaM cAM P → AM P + P DE1CaM
cAM P + P DE4 ↔ P DE4 cAM P → AM P + P DE4
P KA + cAM P ↔ P KAcAM P 2
P KAcAM P 2 + cAM P ↔ P KAcAM P 4
P KAr + P KAc ↔ P KAcAM P 4
Table 1. Reactions in the cAMP pathway
19
20
21
22
23
24
25
26
27
28
Ca4CaM + P P 2B ↔ P P 2BCa4CaM
P P 2BCa2CaM + Ca ↔ P P 2BCa4CaM
CaM + P P 2B ↔ P P 2BCaM
Ca2CaM + P P 2B ↔ P P 2BCa2CaM
P P 2BCaM + Ca ↔ P P 2BCa2CaM
CaM + Ca ↔ Ca2CaM
Ca2CaM + Ca ↔ Ca4CaM
Ca4CaM + CaM KII ↔ CaM KIICa4CaM
CaM KIICa4CaM → CaM KIIpCa4CaM
CaM KIIpCa4CaM → CaM KIICa4CaM
kon
1
0.1
1
1
0.006
0.006
0.1
0.00075
koff
0.3
10
3
0.3
0.91
9.1
1000
0.1
kcat
0.005
0.015
Table 2. Reactions in the Ca pathway
kon
0.00111
0.0006
6e-005
0.00333
koff
10
0.001
0.0003
10
kcat
20
10
100
0.0385
50
0.000128
0.001
0.0192
0.261
0.9
25
28.46
6e-005
0.1
0.0046
0.02
2.6e-005
3.46e-005
0.00102
0.131
1
44
72
0.006
0.06
0.0048
14.23
11
18
5
29
30
31
32
33
34
35
36
37
38
39
40
DARP P 32 + P KAc ↔ DARP P 32P KAc → pT hr34 + P KAc
P P 2A + P KAc ↔ P KAcP P 2A → P P 2Ap + P KAc
P P 2Ap → P P 2A
pT hr34 + P P 2BCa4CaM ↔ pT hr34P P 2B →
DARP P 32 + P P 2BCa4CaM
pT hr34 + P P 2A ↔ pT hr34P P 2A → DARP P 32 + P P 2A
DARP P 32 + Cdk5 ↔ DARP P 32Cdk5 → pT hr75 + Cdk5
pT hr75 + P KAc ↔ pT hr75P Kc
pT hr75 + P P 2Ap ↔ pT hr75P P 2Ap → DARP P 32 + P P 2Ap
pT hr75 + P P 2A ↔ pT hr75P P 2A → DARP P 32 + P P 2A
1P P 2A + 4Ca ↔ 1P P 2Ac
pT hr75 + P P 2Ac ↔ pT hr75P P 2Ac → DARP P 32 + P P 2Ac
AM P → AT P
kon
0.0027
0.0025
koff
8
0.3
kcat
2
0.1
0.004
0.001
0.0001
0.00045
0.00037
0.0004
0.0001
7.72e-012
0.0004
2
2
2
1
12
6.4
0.01
12
0.5
0.5
0.5
3
1.6
3
10
Table 3. DARPP-32 reactions
D1R
Gabc
AC5
AT P
500
3000
2500
2e+006
CaMKII
DARP P 32
P P 2A
P P 2B
20000
50000
2000
4000
P DE1
P DE4
P KA
CaM
4000
2000
1200
10000
PP1
Cdk5
Ca
Da
5000
1800
1000
5000
Table 4. Initial Concentrations
PSF Analysis
In order to set up source-target functions, we need to select input nodes from the available
species nodes. In this example, we used Da (dopamine as ligand for the D1 receptor) and
Ca (extracellular calcium that diffuses through ion channels in the membrane). We use
input concentrations over a specified range (e.g., between 60nM and 5µM for Da), sample
over the range with e. g. n=20 steps, and use the differential equation implementation of
the system to calculate the output values for all species for each sampling step. Because of
the conservation of molecules, all species reach steady-state after a sufficient period of time.
We define steady-state pragmatically by relative change of less than 2% over 100s. We also
use the established terminology of EC10, EC50, EC90 etc. to indicate 10, 50 or 90 % of
steady-state concentration value. Additionally, we calculate the delay in reaching steadystate. We store input-target concentration mappings in a vector (single-input system), or a
matrix (multiple-input system). We fit the vectors with hyperbolic or linear functions, using
standard techniques in Matlab (fminsearch, [25]). In this way we derive parameters which
can be analyzed and used instead of the explicit vectors. (In this paper, the fitting is only
done for single-input systems, multiple-input systems require different techniques.)
All information on source-target transfer functions for the complete, complex signaling
system (’systemic psf’) can be stored in a static data structure. For each species, it contains
its concentration range, and for each reaction, it contains the parameters of the functional
fit. We gain the possibility to regard any species as source and any other species as target
(they may be coupled by an arbitrary number of reactions) and obtain a systemic psf as the
transfer function between them. This representation allows to analyze the complex signaling
system by its parts, i.e. as a set of matrices or vectors, which is the main achievement
6
relative to the ODE dynamical system. In addition, dynamical simulation with appropriate
update times may be realized by the psf representation alone, i.e. the psf simulation is not
in itself atemporal, but only discrete and fairly slow.
We visualize the data structure as a bipartite graph, and label it with the calculated
numeric values. Each species node is labeled with its attainable concentration range given the
input range. For complex formation reactions, we show both [A] → [AB] and [B] → [AB].
For enzymatic reactions we show [E] as the source and [A*] as the target ([E] → [A∗]). The
result is a labeled bipartite graph, called a ’weighted dynamic graph’.
Results
Elementary Biochemical Reactions
We want to represent a biochemical reaction by a time-independent signal transfer function,
such that y = f (x) for two species x, y. We do this by designating a source species x and
then calculating the steady-state value for another species, the target species y, for any value
of x, given the differential equations for the biochemical reaction. For complex formation
[A] + [B] ↔ [AB]
where the total concentrations for [A] and [B] and kinetic rate parameters kon , koff (with
kof f
kon ) are given, the differential equations are:
Kd =
dxdt(A) = koff [AB] − kon [A][B]
dxdt(B) = koff [AB] − kon [A][B]
dxdt(AB) = −koff [AB] + kon [A][B]
We may now calculate the concentration values f (x) = y for a target species [AB] given
a range of input values for x, e.g. the source species [A]. (Fig. 1A).
In this way we separate the calculation of the signal response magnitude, i. e. the steadystate concentration, from the calculation of the time until a steady-state value is reached,
the delay. For different x, f (x) will be reached after a variable delay (Fig. 1B).
With some modification, the same transformation applies to enzymatic reactions. The
f
kinetic rate parameters are Kd = kof
kon (for ↔) and kcat (for →).
[A] + [E1 ] ↔ [AE1 ] → [A∗ ] + [E1 ]
Here it is required that the enzymatic reaction is reversible, i.e. a reaction
[A∗ ] → [A]
exists. (For instance,
[A∗ ] + [E2 ] ↔ [A∗ E2 ] → [A] + [E2 ]
is a reaction that reverses [A*]. ) The differential equations, with kcat2 for [A∗ ] → [A], are:
7
dxdt(A) = koff [AE1 ] − kon [A][E1 ] + kcat2 [A∗]
dxdt(E1 ) = (koff + kcat )[AE1 ] − kon [A] ∗ [E1 ]
dxdt(A∗) = kcat [AE1 ] − kcat2 [A∗]
dxdt(AE1 ) = kon [A][E1 ] − (koff + kcat )[AE1 ]
Given concentrations for E1 , A and kinetic rate parameters Kd, kcat and kcat2 , we may
now derive a function with x as the source species [E1 ] and y as the target species [A∗]
(Fig. 1C).
In both cases the resulting curve can be fitted by a saturating hyperbolic function.
y = f (x) = ymax − (
ymax − ymin
)
1 + ( Cx )n
Here ymin , the baseline concentration, is usually set to 0.
If we choose [A] as the target of [E1], we get a negative slope psf.
y = f (x) = ymin − (
ymin − ymax
)
1 + ( Cx )n
We call this function the elementary protein signaling function or elementary psf.
This function is somewhat related to a Hill equation [26, 27]. A Hill equation is a function
fitted to an experimental measurement to derive a dose-response relationship, comparable
to the psf. The Hill equation allows to calculate a fractional concentration θ for the target
(e.g. a receptor-ligand complex) from the source concentration [L], given Kd, and fitting a
parameter n for the steepness of the curve.
θ=
[L]n
Kd + [L]n
The concentration of the other compound of the complex is not used (assumed large), and
the absolute magnitude of the target is not calculated. An equivalent for enzymatic reactions
is not defined. The parameter n allows to measure the effect of competing binding reactions
(n=1 if none are present), which in our terminology translates into a systemic psf with
multiple binding partners for a single target compound. Systemic psfs are a more general
concept than Hill equations, but they relate to the same type of data, namely dose-response
functions in steady-state.
We have seen that signal transmission strength is uniformly characterized by saturating
hyperbolic functions. This means that it is highest for low x and diminishes as x increases
(Fig. 1D). For instance, in Fig. 1D, a 100% signal increase leads to 100%, 18% or only
6% increase in the target depending on the source concentration. For enzymatic reactions,
absolute concentration changes have different effects for sources and targets of a signaling interaction. Signal transmission strength depends on the absolute concentration of the
source, the target concentration is irrelevant. This is an important observation, since protein
signaling systems are subject to long-term regulation of concentrations. In the context of
disease states or other sources of protein expression up-/downregulation, independence of
transmission strength from target concentration may be an important conservative property.
8
Signal transmission is strongest if a source species is expressed at a low concentration. We
need to bear in mind however, that reaction velocity operates inversely to signal transmission
strength: a low source concentration means a slow reaction (Fig. 1B). A functioning signaling
system would therefore have to use an intermediate range to maximize signal transmission
within time constraints. Our analysis opens a new way of analysis for a signaling system:
Optimization techniques could find a best source range for both time and signal transmission
constraints.
Systemic PSF Analysis
A source-target psf can be derived for any pair of species in a complex biochemical reaction
system. For a complex system, or set of equations, we define a set of input nodes, and
compute the output values for each possible input configuration. The analysis gives us the
output concentration range (notwithstanding transients in a dynamic context, s. below)
for each species, as well as a (fitted) function, or matrix of input-output correspondences.
A biochemical reaction will produce a different psf, when it is elementary or when it is
embedded in a context, where the participants of the elementary reaction also participate in
other reactions. This is true for both protein complex formation and enzymatic reactions.
We therefore call source-target functions ’systemic psfs’, when they are derived from the
context of a specific signaling system.
We have provided this analysis for the example system. We show the concentration
ranges and the signal transmission functions for the whole system [3] as a weighted dynamic
graph for Da as a single input (Fig. 2). We label each species node with its concentration
range, determine source and target species nodes for each reaction node, and provide fits for
the systemic psf, the transfer function that characterizes each reaction. We see from (Fig. 2)
that a number of systemic psfs can be fitted well with a linear function (y = mx + b),
showing that systemic psfs sometimes consist only of a short section from a full mapping of
concentration values. Also, many species have only small concentration ranges, which means
they don’t have much response to Da input.
It is an obvious advantage of the psf analysis that we are able to dissect the complex
system and extract local properties, such as concentration ranges of individual species, and
transfer functions for individual reactions under input stimulation. This allows to critically
analyze a model, compare these properties with biological data, and adjust or improve the
model in a detailed manner.
In Fig. 3, the concentration ranges for some target species are given. We see, for instance, that among DARPP-32 phosphorylation variants, pThr75 is always more abundant
that pThr34, by an order of magnitude. This is an example of a high-level property, which
could be related to biological data. As another example, we notice that the active receptor
conformation (Da-D1R) remains below 160nM even under stimulation with 1µM Da and
more. With a D1R total concentration of 500nM, we could adjust the ligand binding coefficient to produce more or less active receptors. Finally, the analysis shows a very low maximal
PKAc level (12nM) in spite of a total PKA concentration of 1.2µM. In the original model
[3], blind parameter adjustment has probably generated a very low level of PKAc in order
to achieve high signal transmission for phosphorylation of the target species pThr34, which
is experimentally required, but which could be achieved in other ways (e.g. PP2B) as well.
9
With our analysis, properties of individual species become apparent, and they can be compared to biological data, tested and adjusted on a localized basis. Even more interestingly,
we could look for principles of ’rational system design’, for instance question the transmission
of a seven-fold increase of cAMP in the µM range to a maximal three-fold increase in the
10’s of nM for PKAc, and analyze given biological systems from this perspective.
In addition to the concentration ranges, we also have access to the functional mapping
between species in the model. The systemic psfs, like the elementary psfs, are stored as
vectors, which are matched by functional parameters. The advantage of the psf analysis is
that we can probe a complex system on a single reaction level because the influence of the
cellular context is encoded in the systemic psf. Thus we can compare the elementary psf with
its transformation as a systemic psf for individual reactions. Fig. 4A shows elementary and
systemic psfs for G-protein activation of AC5 and the calcium-activated complex AC5Ca. We
see that the systemic psfs are somewhat deflected, compared to the elementary psf, which is
what we expect from the parallel activation of AC5Ca and AC5 by the same species. We may
specify a desired psf using only functional parameters, and adjust elementary parameters to
match the psf (Fig. 4B). A local change to the Kd binding coefficient between AC5Ca and
GoaGTP allows a change in the systemic function. Since other systemic psfs may be affected
by such a change - this can be detected by re-computing the weighted dynamic graph - more
adjustments of elementary rate parameters may be indicated, possibly by an iterative process
(cf. [28]).
Sampling for multiple inputs yields a transfer function matrix, which can be analyzed
for dependence of the target concentration on each input separately. This can be done
by standard matrix analysis such as principal component analysis (PCA). For our example, we show how species which are poised to integrate signals from two different sources
do this under the numeric conditions (Fig. 5). For cAMP production (AC5), we find that
AC5GoaGTP is dependent only on Da (Fig. 5A), AC5Ca is only dependent on Ca (Fig. 5B),
while AC5CaGoaGTP is almost not activated at all (Fig. 5C). Even though a link of reactions (Ca-AC5Ca-AC5CaGoaGTP-cAMP) exists, signal integration of Ca and Da on AC5
fails because of the weak transmission from GoaGTP to AC5Ca. Signal integration between Ca and Da occurs for cAMP degradation by calcium-dependent calmodulin regulation
of PDE1. PP2A with the two variants (calcium-activated) PP2Ac and (PKAc-activated)
PP2Ap is another potential source of signal integration (Fig. 5D,E,F). The psf analysis
shows when signal integration occurs (here: Da having influence on PP2Ac), and when this
effect is negligible (here: Ca not having influence on PP2Ap). This may now be studied
for correspondence with the biological situation. These results emphasize the necessity for
numeric analysis of input-dependence, beyond the mere existence of links.
Systemic Delay and State-change Dynamics
We would like to be able to use systemic psfs with their simple and transparent mathematical
structure for dynamical simulations. This allows direct experimental testing and fitting by
time series measurements beyond dose-response relationships. In order to do this, we need to
compute the systemic delays, i.e. the reaction time until a steady state is formed. Then we
can build a state-change dynamical model from systemic psfs alone, using the appropriate
delays for the input and the update of a system state.
10
Species
DaD1R
AC5GoaGTP
AC5CaGoaGTP
cAMP
PKAc
pThr34
PP2Ap
PP2Ac
pThr75
0.06 → 0.5µM
0.06 → 4.5µM
0.5 → 0.06µM
4.5 → 0.06µM
3.6
7.8
8.1
7.8
251
288
511
600
493
0.8
2.4
2.4
3.1
164
200
387
483
359
10.1
19
19.5
18.1
347
369
706
756
756
10
20
26
18
300
307
683
717
755
Table 5. Delays: For near instantaneous input signal Da at shown concentrations, the
table shows delays (in s) to reach EC5/EC95 for target species. Decrease is usually slower
than increase, due to the asymmetry of kon /koff binding parameters. Delay times are
sensitive to the absolute size of the signal, with delays being faster for larger signals.
Systemic delays depend on the absolute size of the signal and also the direction (increase/decrease) of signaling. Delays for species in the example system in response to input
are shown in Table 5. For the computation of target concentrations, we only need a ratio
kcat
such as kon
= Kd (binding coefficient) or kcat2
for forward and backward enzymatic reackoff
tions. For the delays, the difference between kon and koff or kcat and kcat2 defines reaction
times for synthesis and degradation. Therefore, delay computations are fairly complex, but
the results are often within a fairly narrow range for each reaction (Table 5). For discrete
state-change simulations we may use maximal delays for each species.
From a biological perspective, this table provides an important test on the validity of
the model. In many cases, systemic delays can be measured. For instance, the delay for
PKAc at 150-250s rather than 30-60s, as measured in [29] (cf. [9]), seems large and may be
an indication for a revision of the underlying parameters. From the theoretical perspective,
this system seems to operate on separate time scales: 1-10s, 150-300s and 450-600s. Such a
separation of reactions by their characteristic delay times is interesting, since it could lead
to simulation models with different discrete time scales. Here we may calculate psf values
for fast species with 10s time resolution, for intermediate species with 300s time resolution,
and for slow species with 600s time resolution, i.e. system update time for state changes
(Fig. 6A,B). It is an empirical question, whether separate time scales rather than a continuum
of delay values will prove to be an organizing principle in protein signaling systems [30]. A
general study, for instance, using models from the BioModels Database [19], might give
answers to this question. Time scale separation may provide a conservative property of a
signaling system against fluctuations of concentrations. If total concentrations in the system
change, e.g. by protein expression up- or down regulation, miRNA interaction, or diffusional
processes across compartments, the relevant interactions will continue within each time slice.
Concerted regulation of protein expression levels may set a clock for the rapidity of signal
transduction.
Systemic dynamics, in contrast to elementary reaction dynamics, need not follow a hyperbolic curve. If there are feedbacks in the system, the dynamics may contain transients,
11
i.e. the concentration may be higher or lower before it settles into its steady-state value [31].
The dynamic response of target species to input are shown in Fig. 6A,B. For a species without a transient response, the actual value of a species at a shorter delay is always bounded
by the steady-state value, and all possible concentrations in a continuous-time dynamical
system are bounded by the psf concentration range [30]. However, if there are transients,
a psf-based dynamical simulation will miss these transients and plot a simplified trajectory.
This means that results from a psf analysis with slow inputs cannot be extrapolated to much
faster input dynamics - in contrast to continuous-timed dynamical systems where arbitrary
time units can be chosen. This restriction may capture a biological reality: steady-state
behavior provides the framework and may operate according to rules and principles which
are separate from the effects of short term fluctuations.
The psf system allows to generate a dynamical system as a sequence of states defined by
fluctuations of input. Fig. 6C,D shows an overlay of a differential equation simulation and
psf state change simulation for a sequence of inputs with 10s duration. Accordingly, the psf
approximation is excellent for all species with a delay time of < 10s. If we plot psf values for
slow species at intervals corresponding to their maximal delays, we may linearly interpolate
between points, and in this case achieve a good approximation of the continuous-time model.
A psf-based dynamical system is an important tool in order to generate a time-series
simulation from a calculated model system for comparison with experimental data. The psf
system utilizes parameters which are uniform and have linear error ranges (cf. Fig. 1A,C),
and therefore should improve interaction of the model with the experimental reality. The
psf model will also allow to predict the optimal stimulation times for different inputs such
that responses can be measured in steady-state.
Computing Input-dependent Modularity
Since we are able to define signal transmission capacity, we have a tool to investigate modularity of a signaling system. As species saturate or return to basal levels, they act as inactive
links, i.e. they are stuck at the same concentration value, and cannot transmit further increases or decreases of inputs. We hypothesize that this effect is actually important in many
protein signaling systems. We may define an inactive connection as a species node which has
only limited (e.g. < 10%) signal transmission capacity. The interconnectedness of a system
is then proportional to the number of inactive connections, and a module is a part of the
system with few or no active connections to the rest of the system.
In the following we discuss the activation/inactivation of links with respect to input
increases. I.e. given a certain level of extracellular signaling, what happens if this level is
raised and then kept at the higher level for some time, sufficient for the system to settle
into a new steady state? Which species nodes will respond to the increase and transmit
it to downstream targets, and which species will become saturated and only respond with
their saturated value? It is also clear that species which have become saturated (inactive)
will not respond to fast extracellular signals anymore. This analysis is therefore useful
both for the steady-state context and for understanding fast input fluctuations. There is
an input level for each target species, where the species ceases to be responsive to further
input increases. If that input level has been reached, the species can be considered to have
become an inactive connection, i.e. a node which does not transmit signals. We may define
12
systemic psfs for ’input-target psfs’ from the input to any target species (Fig. 7). We notice
how number of steps in the computation of a species concentration relates to a lower cutoff value for signal transmission (e.g. DaD1R, AC5GoaGTP, cAMP vs. PKAc, pThr34,
pThr75). In other words, earlier steps in the sequence saturate at a higher input level than
later steps. Two parallel targets (pThr34, PP2Ap) of an intermediate step (PKAc) may
saturate at very different input levels (∼ 3µM for pThr34 vs. 1.5µM for PP2Ap). This
mechanism demonstrates the effect of a sequence of saturating functions, and constitutes a
general principle in the construction of a signal transduction system.
The model allows to study whether a node responds to a specific input with any change of
steady-state. In Fig. 8 and supplemental figures 1, 2, we show modularization of the system
under various Da and Ca input conditions. With Da input and high Ca (Figure 8B), species
which are proximal to Da input, the receptor-ligand complex DaD1R, the signaling complex
through G proteins and adenylyl cyclase AC5, as well as cAMP, are most responsive to Da
over a large range of input. Species in the ’integration zone’ between Da and Ca, such as
DARPP-32, PP2A cease responding to Da increases at lower levels and become inactive links
at higher levels. Species in the system with no significant change in concentration at any
input level are for the most part proximal to Ca instead of Da, and thus highlight modularity
among pathways. In this case, we see that Da inputs are transmitted to distant targets only
up to an intermediate range and that there are a significant number of species which do not
react to Da at all. Above that range, even though closely coupled targets still respond to the
input, the signal increase is not registered beyond cAMP production and synthesis. With Ca
low (supplemental Figure 1) there is widespread responsivity to Da up to 1-2µM and only a
few of the calcium-related species do not respond at all. The Da signal is therefore able to
influence the calcium-related pathway, provided calcium is low.
When Ca is the input (Fig. 8A), the calcium-responsive proteins like calmodulin, CaMKII,
calcium-activated PP2B (calcineurin) transmit signals, while the GPCR pathway remains
almost completely unresponsive. There is some signal integration with calmodulin-activated
PDE1 for cAMP. Other than that, we see that PP2Ac and the pThr34 variant of DARPP-32
respond strongest to calcium while PP2Ap and the pThr75 variant of DARPP-32 have less
or no responsiveness to calcium. With Da low, as in supplemental Figure 2, again, most
of the GPCR-related species do not respond to Ca at all, or cease responding at 10-25% of
maximal Ca (1-4µM). However Ca-related species like calmodulin, CamkII, PP2B remain
responsive. The difference between the high and low Da condition is small. Here the few
existing links from Ca to GPCR (such as Ca regulation of AC5) are not strong enough to
influence the GPCR pathway significantly in any condition. In this case, it is not just the
saturation that matters, the input signal has limited reach in influencing distant species in
general.
We are analyzing a biological system with two ’pathways’, the cAMP and the calcium
pathway, which are cross-linked in a convergence zone of species which are influenced by both
pathways. It is remarkable how clearly three different modules appear: the GPCR/cAMP
pathway, the Ca pathway, and the signal integration zone.
There is also a general observation to be made about signal transmission in a protein
signaling system: Signal integration is strongest when inputs are low. This is a direct consequence of the effect of coupling saturating nodes. It means that there are few saturating
species in the system which impose modularity, and signals spread further. Widespread
13
interconnectedness is only possible at low input levels. Because activation curves for biochemical reactions are mostly uniform hyperbolic (saturating), a stronger modularization
with many inactive links results from higher input levels. This may also have implications
on transient responses. For instance, phosphatase and kinase response often differs with
kinases being saturated only at higher input levels. If and where that is the case, we may
observe transient responses that only reflect kinase activity, since phosphatases are saturated
and do not participate in signaling, they do not add or subtract and in this way obscure the
kinase signal. This shows that the steady-state input-response system may work well as the
framework wherein fluctuating signaling operates.
Discussion
Continuous dynamical systems - systems that use change over time as a system primitive - are
notoriously difficult to analyze and may not be the best choice of a tool for signal transduction
systems of moderate or large size. Steady-state matrix computations are simple and fast,
scale well to very large sizes, and offer multiple opportunities for analysis. By calculating
transfer functions from a systemic dynamical model, we also gain the opportunity to extract
and analyze parts of the system. This may help in creating re-usable system parts. This
paper demonstrated a transformation of a mass-action kinetic biochemical reaction model
implemented by a set of differential equations into an input-response transfer function model.
The transformation is done by calculating steady-state concentrations for each species in
response to a range of input values, and then analyzing the resulting vectors (matrices) as
the basis of a transfer function (’psf’). For small, toy-like networks of few components, such
dose-response relationships have been investigated by [32]. They analyse kinetic models in
the same way, however, they don’t make a distinction between parameters in elementary
interactions, and the actual parameters in a systemic context. They also do not address
the question of temporal embedding of the dose-response relations. In our approach, we
use dose-response functions, similar to Hill functions, but we are extending the concept. By
using rectangular signals, i.e. constant signaling levels, we calculate the response as the
steady-state value. In addition, we calculate the time to steady-state from the underlying
dynamical model. Only because of this additional computation can one attempt to create
a discrete dynamical model – something that is not within the purview of a Hill equation
model. Hill equations attempt to fit or create systemic parameters, which are different from
elementary parameters, i.e. they do recognize the dependence of the transfer function on
the systemic context. The psf model is an approach of making Hill equations (dose-response
relations) work in large-scale modeling.
There are numerous attempts to simplify dynamical models in the temporal domain
by creating hybrid models (e.g. [33]). Sometimes slow interactions are regarded as constant, and only fast interactions are dynamically modeled. Sometimes, fast interactions are
replaced by time-independent values and only slow interactions are dynamically modeled
(many examples, for instance [34]). There are also attempts to replace an ODE model by a
delay-differential equation (DDE) model, i.e. to compute and use explicit time delays and
eliminate many intermediate species, simplifying the size of the model [35].
In our approach, time and concentration are regarded as separate, which makes it different
14
from any hybrid approach. The main restriction is the assumption of a constant signaling
level over periods of time sufficient to induce steady-state. The approach is therefore best
suited to check the limiting conditions of a dynamical model, e.g. in drug development
applications, where multiple dose-response relations derived from the model can be crosschecked and used to locally improve the model. However, the simple, atemporal transfer
functions can also be applied directly: (a) in experimental settings where signal duration
can be controlled, and (b) in physiological settings, when fluctuations occur around different
mean values, and we model the step changes for the mean extracellular signaling level,
but not the short-term fluctuations. It could be shown that psfs are sufficient to create
discrete time dynamical models, with certain restrictions on fast dynamical inputs below
the time resolution of the system. The primary focus of the analysis was on single-input
systems, where signaling functions can be matched by parameterized hyperbolic functions.
We showed how the analysis can be extended to multiple-input systems by computing and
analyzing psf transfer matrices. The psf functions allow to cut through the complexity of
the model and examine interactions in a localized way. If continuous dynamical models are
being used with the goal of adequately simulating cellular processes, this kind of analysis is
an indispensable tool to check for the consequences of the modeling assumptions in terms
of dose-response relationships. The analytical tools of the psf approach may also be used to
systematically investigate the effect of localized changes to the system [36, 28], and to offer
local corrections to the complex system in a transparent way.
The extraction of input-target psfs with characteristic hyperbolic saturating properties
allows to determine input level dependent inactivation of a species node. Accordingly, we
can define the limits of signal transmission by the distribution of inactive nodes. In the
example system, a strong distinction of calcium- and cAMP-dependent pathways and a
signal integration zone were revealed. We could see that at low levels of input, widespread
interactions are possible, while at higher levels of input, many species enter into a state of a
constant function value and become inactive links. This corresponds to biological results and
expectations, and provides a foundation for the concept of pathways in signal transduction.
For transient responses, the steady-state input-response system provides the boundary values
to which the system eventually settles. This is especially interesting if we have saturating
response levels, which are unresponsive to further signaling (e.g. phosphatases), and allow
transient signals (e.g. kinases) to emerge.
Signal transduction may also be analyzed from the perspective of rational system design.
Such work is still in its infancy. We may for instance investigate the effect of negative
feedback links on concentration ranges and times to steady state. Another question would
be the optimization of a signal transduction system for the trade-off between speed and
efficacy of signal transmission. With this choice of model, many new questions can be
raised, and old problems like parameter dependency, modularity or signal integration can be
addressed in a novel way.
References
1. Bhalla US, Iyengar R (1999) Emergent Properties of Networks of Biological Signaling
Pathways. Science 283: 381–387.
15
2. Gilbert D, Fuss H, Gu X, Orton R, Robinson S, et al. (2006) Computational methodologies for modelling, analysis and simulation of signalling networks. Briefings in
Bioinformatics 7: 339–53.
3. Lindskog M, Kim M, Wikström MA, Blackwell KT, Kotaleski JH (2006) Transient
calcium and dopamine increase PKA activity and DARPP-32 phosphorylation. PLoS
Computational Biology 2: e119.
4. Svenningsson P, Nishi A, Fisone G, Girault J, Nairn A, et al. (2004) DARPP-32: An
integrator of neurotransmission. Annual Review of Pharmacology and Toxicology 44:
269–296.
5. Cepeda C, Levine MS (2006) Where do you think you are going? The NMDA-D1 receptor trap. Science’s STKE: signal transduction knowledge environment 2006: pe20.
6. Yger M, Girault JA (2011) DARPP-32, Jack of All Trades. . . Master of Which? Frontiers in Behavioral Neuroscience 5: 56.
7. Kotaleski JH, Blackwell KT (2010) Modelling the molecular mechanisms of synaptic
plasticity using systems biology approaches. Nature Reviews Neuroscience 11: 239–51.
8. Stefan MI, Edelstein SJ, Le Novere N (2008) An allosteric model of calmodulin explains
differential activation of PP2B and CaMKII. Proceedings of the National Academy
of Sciences of the United States of America 105: 10768–73.
9. Kim M, Huang T, Abel T, Blackwell KT (2010) Temporal sensitivity of protein kinase
a activation in late-phase long term potentiation. PLoS Computational Biology 6(2):
e1000691.
10. Pepke S, Kinzer-Ursem T, Mihalas S, Kennedy MB (2010) A dynamic model of interactions of Ca2+, calmodulin, and catalytic subunits of Ca2+/calmodulin-dependent
protein kinase II. PLoS Computational Biology 6: e1000675.
11. Manninen T, Hituri K, Kotaleski JH, Blackwell KT, Linne ML (2010) Postsynaptic
signal transduction models for long-term potentiation and depression. Frontiers in
computational neuroscience 4: 152.
12. Nakano T, Doi T, Yoshimoto J, Doya K (2010) A kinetic model of dopamineand calcium-dependent striatal synaptic plasticity. PLoS Computational Biology 6:
e1000670.
13. Fernandez E, Schiappa R, Girault JA, Le Novere N (2005) DARPP-32 is a robust
integrator of dopamine and glutamate signals. PLoS Computational Biology 2(12):
e176.
14. Le Novere N, Li L, Girault JA (2008) DARPP-32: molecular integration of phosphorylation potential. Cellular and Molecular Life Sciences: CMLS 65: 2125–7.
16
15. Qi Z, Miller GW, Voit EO (2008) Computational systems analysis of dopamine
metabolism. PloS One 3: e2444.
16. Qi Z, Miller GW, Voit EO (2010) The internal state of medium spiny neurons varies
in response to different input signals. BMC Systems Biology 4: 26.
17. Oliveira RF, Kim M, Blackwell KT (2012) Subcellular Location of PKA Controls
Striatal Plasticity: Stochastic Simulations in Spiny Dendrites. PLoS Computational
Biology 8: e1002383.
18. Barbano PE, Spivak M, Flajolet M, Nairn AC, Greengard P, et al. (2007) A mathematical tool for exploring the dynamics of biological networks. Proceedings of the
National Academy of Sciences of the United States of America 104: 19169–74.
19. Li C, Donizelli M, Rodriguez N, Dharuri H, Endler L, et al. (2010) BioModels
Database: An enhanced, curated and annotated resource for published quantitative
kinetic models. BMC Systems Biology 4: 92.
20. Hughey JJ, Lee TK, Covert MW (2010) Computational Modeling of Mammalian Signaling Networks. Wiley Interdiscip Rev Syst Biol Med 2(2): 194–209.
21. Wu J, Vidakovic B, Voit EO (2011) Constructing stochastic models from deterministic
process equations by propensity adjustment. BMC Systems Biology 5: 187.
22. Erhard F, Friedel CC, Zimmer R (2008) FERN - a Java framework for stochastic
simulation and evaluation of reaction networks. BMC Bioinformatics 9: 356.
23. Deng J, Feinberg M, Jones C, Nachman A (2011) On the steady states of weakly
reversible chemical reaction networks. Arxiv preprint arXiv:11112386v2 : 1–20.
24. Bandara S, Schlöder J, Eils R, Bock H, Meyer T (2009) Optimal experimental design
for parameter estimation of a cell signaling model. PLoS Computational Biology 5:
e1000558.
25. Lagarias J, Reeds J, Wright M, Wright P (1998) Convergence properties of the NelderMead simplex method in low dimensions. SIAM Journal of Optimization 9: 112–147.
26. Kraeutler MJ, Soltis AR, Saucerman JJ (2010) Modeling cardiac β-adrenergic signaling with normalized-Hill differential equations: comparison with a biochemical model.
BMC Systems Biology 4: 157.
27. Yu RC, Resnekov O, Abola AP, Andrews SS, Kirsten R, et al. (2010) The Alpha
Project: a model system for systems biology research. IET Syst Biol 2: 222–233.
28. Maslov S, Sneppen K, Ispolatov I (2007) Spreading out of perturbations in reversible
reaction networks. New Journal of Physics 9: 273.
29. Woo NH, Duffy SN, Abel T, Nguyen PV (2003) Temporal spacing of synaptic stimulation critically modulates the dependence of LTP on cyclic AMP-dependent protein
kinase. Hippocampus 13(2): 293–300.
17
30. Purvis JE, Radhakrishnan R, Diamond SL (2009) Steady-state kinetic modeling constrains cellular resting states and dynamic behavior. PLoS Computational Biology 5:
e1000298.
31. Theis F, Bohl S, Klingmueller U (2011) Theoretical analysis of time-to-peak responses
in biological reaction networks. Bulletin of Mathematical Biology 73: 978–1003.
32. Tyson JJ, Chen KC, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics
of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 15(2): 221–231.
33. Eriksson O, Brinne B, Zhou Y, Björkegren J, Tegnér J (2009) Deconstructing the core
dynamics from a complex time-lagged regulatory biological circuit. IET Syst Biol 3(2):
113–29.
34. Saucerman JJ, Brunton LL, Michailova AP, McCulloch AD (2003) Modeling betaadrenergic control of cardiac myocyte contractility in silico. J Biol Chem 278(48):
47997–8003.
35. Srividhya J, Gopinathan MS, Schnell S. (2007) The effects of time delays in a
phosphorylation-dephosphorylation pathway. Biophys Chem 125(2–3): 286–97.
36. Maslov S, Ispolatov I (2007) Propagation of large concentration changes in reversible
protein-binding networks. Proceedings of the National Academy of Sciences of the
United States of America 104: 13655–60.
18
A
Kd=200
Kd=500
6000
200
B
7000
180
160
5000
140
C
C
1/2 ymax
C
9.86 4.37
[AB](nM)
[AB](nM)
Kd=2000
4000
0.929
3000
120
100
80
2000
200nM
1000 nM
2000 nM
5000 nM
7000 nM
60
n
f(x)=ymax−ymax/(1+(x/C) )
40
1000
20
0
0
500
1000
1500
2000
2500
[A](nM)
3000
3500
4000
4500
5000
D
140
C=349
2
3
4
5
C=1117
kcat2/3
A* [ nM]
C=1038
80
1600
EC95
500
1000
+6%
1200
C=1020
100
1800
1400
120
A*[nM]
1
sec
160
C
0
0
+48%
1000
EC50
100
800
200
60
600
C=2063
kcat*2
0
+100%
1
200
kcat2*2
0
100
200
)
400
kcat/3
f(x)=200 − 200/(1+(x/C) )
20
1.4
f(x)=1800 − 1800/(1+(x/93)
40
300
400
500
E1[nM]
600
700
800
900
1000
0
EC5
17 34
0
100
200
300
400
500
E1[ nM]
600
700
800
900
1000
Figure 1. Properties of psf functions. A. Complex formation: Psfs were generated for
[A] + [B] ↔ [AB] with Kd = 200, 500, 2000, [A]0 = 10nM − 10µM; [B]0 = 7µM, and fitted
with the saturating hyperbolic function shown in the figure. For an elementary reaction,
C = Kd. All other curves were generated only by varying C, where ymax = [B]0 and n=1.
The slope at C indicates the signal transmission strength. B. Dynamics of complex
formation: [A] + [B] ↔ [AB] with Kd = 200, [B]0 = 200nM; [A]0 = 200nM − 7µM.
Increasing concentrations for the source species [A]0 speeds up the reaction. C. Enzymatic
Reactions with Reverse Reaction: Example reaction with [A]0 = 200, kon =0.00026; koff =1.5;
kcat =30.4; kcat2 =0.26 (shown in red) and variations (shown in green, blue). Both the
forward reaction (kcat ) and reverse reaction (kcat2 ) parameters are varied by 30%–200%.
Variability in signal transmission is expressed by the fitted parameter C (half-maximal
activation) in a uniform way. D. Signal Transmission Strength: A 100% increase of input
yields diminishing increases at higher concentrations. Signals are shown at EC5, EC50 and
EC95.
19
Da
84...4797
D1R
254...402
y_max=425 C=7460 n=1
y_max=350 C=166 n=1
DaD1R
6...167
AC5
1907...2235
y_max=25 C=120 n=1
y_max=62 C=2.2 n=0.67
DaD1RGabc
3...15
Ca
64...69
CaM
4795...4874
y_max=565 C=2400 n=2
y_max=625 C=85 n=4
Gbc
98...487
GoaGTP
5...30
GoaGDP
0...0.1
AC5Ca
136...172
y_max=500 C=40 n=4
y_max=22 C=105 n=1
PP2BCaM
2236...2304
y_max=234 C=136 n=1
y_max=435 C=3570 n=1
PDE1
3380...3398
AC5CaGoaGTP
1...3
y_max=2.38e5 C=2570 n=1.2
b=-4930 m=2
y_max=10000 C=30 n=1
PDE4
1420...1881
y_max=127 C=49 n=1.34
y_max=86 C=49 n=1 T
GabcD1R
12...237
b=4000 m=-2
AMP
230...1133
y_max=90 C=1.2e5 n=0.5
y_max=65 C=2 n=0.67
ATP
1.99e6...2e6
y_max=1.09e5 C=137 n=1.24
cAMP
284...1839
b=12500 m=-12
PDE1CaM
528...606
y_max=3.2 C=65 n=4
y_max=7.3 C=2800 n=0.5
PP2BCa2CaM
834...922
y_max=4700 C=150 n=2.5
y_max=2100 C=1300 n=2.5
CaMKII
19648...19696
y_max=2000 C=3.7 n=1.33
y_max=5000 C=12 n=1
y_max=1130 C=5 n=1.25
y_max=5000 C=5500 n=1.1
PP2BCa4CaM
519...620
CaMKIICa4CaM
228...264
y_max=2900 C=630 n=2
y_max=630 C=32 n=1.5
y_max=1300 C=400 n=0.25
y_max=770 C=1250 n=1.33
y_max=1040 C=135 n=1
y_max=1e4 C=3.7 n=2.5
Ca4CaM
1.5...1.8
PKA
57...454
PKAcAMP2
457...614
y_max=6800 C=3700 n=2.5
y_max=5500 C=2 n=1
Ca2CaM
217...232
y_max=1000 C=45 n=1
y_max=1000 C=12 n=1
AC5GoaGTP
9...43
y_max=3000 C=3100 n=1
y_max=1000 C=0.01 n=1
Gabc
2886...2721
y_max=410 C=60 n=2
y_max=260 C=25 n=1
PP2B
1.4...1.5
y_max=3.9e4 C=0.03 n=0.5
PKAr
95...202
Cdk5
233...280
y_max=8300 C=650 n=2.3
y_max=3900 C=30 n=2
y_max=25 C=3.4 n=1
y_max=31 C=6.5 n=1.66
PKAcAMP4
92...484
PP2Ac
9...20
y_max=105 C=23 n=1
y_max=42 C=2300 n=0.75
PKAc
5...11
y_max=15.3 C=1630 n=1.6
y_max=1.3e4 C=70 n=1.3
pThr75
8289...11099
y_max=165 C=660 n=0.35
CaMKIIpCa4CaM
76...88
DARPP32
36160...37417
PP2A
588...988
y_max=2450 C=17 n=0.83
PP2Ap
615...1026
y_max=500 C=540 n=1.4
y_max=1830 C=1310 n=0.27
pThr34
615...1971
y_max=1830 C=1310 n=0.27
pThr75PKAc
19...35
Figure 2. Weighted Dynamic Network View of the Model System. Input
conditions are Ca=8µM and Da=60nM − 5µM (total available concentration). Each
species node is labeled with its steady-state concentration range. Reaction nodes are
labeled by parameters for a hyperbolic fit, or a linear fit if that was sufficient. Complex
formation reactions (red) are labeled for both [A] → [AB] and [B] → [AB] (left to right),
enzymatic reactions (blue) are only labeled for [E] → [A∗]. The low free concentration for
Ca (64-69nM) is a feature of this model, where Kds are set in such a way that most Ca is
bound to calmodulin, and then to other proteins (cAMKII Kinase, PP2B phosphatase and
PDE1), such that all ions are indeed accounted for (some species have 2 or 4 Ca ions
bound). The present analysis allows to critically evaluate the effect of elementray kinetic
parameters on steady-state properties.
20
pThr34
pThr75
PP2Ac
PP2Ap
PKAc
cAMP
AC5GoaGTP
DaD1R
Da_0
0
10
1
10
2
3
10
10
4
10
5
10
[nM]
Figure 3. Ranges of Concentrations in Response to Input (Da). For a number of
relevant target species from the biological model, the range of concentrations for input
(Da) from 100nM to 5µM is shown (Ca - high, 8µM).
21
A
120
100
80
[nM]
AC5GoaGTP
60
ymax=220, C=78, n=1
40
ymax=233, C=126.5, n=1
20
ymax=22, C=100,n=1
0
B
0
10
20
30
GoaGTP [nM]
40
AC5CaGoaGTP
50
60
50
60
120
100
AC5CaGoaGTP
[nM]
80
AC5GoaGTP
60
ymax=220 C=78 n=1
40
ymax=233 C=125 n=1
20
0
0
10
20
30
GoaGTP[nM]
40
Figure 4. Local adjustment of a transfer function. A. Elementary (dashed) and
systemic (continuous) psfs for two targets of GoaGTP. A new systemic psf for
GoaGTP-AC5CaGoaGTP is defined by functional parameter adjustment (black line). B.
Elementary parameters were changed to match the new systemic psf. For AC5CaGoaGTP,
kon =0.0192, koff =25 was adjusted to kon =0.022, koff =1.5, for AC5GoaGTP, kon =0.0385,
koff =50 was adjusted to kon =0.0495, koff =48.5.
22
300
50
40
250
40
30
20
10
AC5CaGoaGTP
50
AC5Ca
AC5GoaGTP
Dose−response
Dose−response
Dose−response
200
150
100
50
5000
0 0
Ca0
1000
2000
3000
4000
5000
5000
Ca0
Da0
A
20
10
0
10000
0
10000
30
0 0
1000
2000
3000
4000
0
10000
5000
8000
5000
6000
4000
3000
4000
2000
2000
Da0
0
Ca0
B
1000
0
Da0
C
Dose−response
Dose−response
1200
Dose−response
200
1200
150
1000
PP2Ac
1000
PP2A
900
800
700
100
50
0
600
500
8000
5000
6000
5000
4000
2000
2000
Ca0
0
Ca0
1000
0
Da0
E
600
200
10000
3000
4000
800
400
−50
10000
400
10000
D
PP2Ap
1100
0 0
1000
2000
3000
4000
5000
5000
Ca0
Da0
0 0
1000
2000
3000
4000
5000
Da0
F
Figure 5. Dependence of target species on Ca and Da inputs. The 2D psf shows
species which segregate to only one of the input pathways (A, B, F), a species which
remains unresponsive (C), and species which show some signal integration in their input
constellation (D, E).
23
A
A
C
500
400
[ %]
300
[nM]
Da
DaD1R
GoaGTP
AC5GoaGTP
AC5CaGoaGTP
cAMP
PKAc
800
PP2Ap
600
pTHr34
cAMP
400
200
Input Da
200
100
0
0
PP2Ac
10
20
30
sec
40
50
0
0
60
B
B
Da
PKAc
pThr34
pThr75
PP2Ac
PP2Ap
[%]
300
150
200
sec
250
300
350
400
30
[nM]
400
100
40
D
500
50
20
DaD1R
10
AC5GoaGTP
200
100
PKAc
0
0
100
200
300
sec
400
500
600
0
0
50
100
150
200
sec
250
300
350
400
Figure 6. Dynamics of target species to Da input. A. Fast species (<10s) have
transients. The psf approximation only calculates the steady-state value. B. For slow
species no transients are apparent. It may take several minutes to reach steady state. C,
D. Dots mark systemic psf values, thick lines are continuous dynamical simulations by
differential equations, thin lines are interpolations between 10s psf values. Red arrows
mark time points for interpolations for slow species like PP2Ac, PP2Ap, pThr34.
24
A
1800
1600
0.178
cAMP
1400
[nM]
1200
pThr34
0.074
1000
PP2Ap
0.064
800
600
400
200
0
0.024 DaD1R
PP2Ac
0.0035
0
500
1000
1500
2000
2500
Da [nM]
3000
3500
4000
4500
5000
0
B
50
45
pThr75
AC5GoaGTP
40
0.00386
35
[nM]
30
25
20
15
AC5CaGoaGTP
0.000423
0.2
5
0
PKAc
0.00071
10
0
500
1000
1500
2000
2500
Da [nM]
3000
3500
4000
4500
5000
0
Figure 7. Systemic psfs for input to target. Shown are EC90 and EC10
concentrations as cut-off thresholds for effective signaling by input and the slope values at
threshold.
25
Da
D1R
A
DaD1R
D1R
B
DaD1R
AC5
Ca
CaM
PP2B
AC5
DaD1RGabc
Gbc
Da
GoaGTP
AC5Ca
PP2BCaM
Ca2CaM
Gbc
GoaGDP
PDE1
PP2BCa2CaM
Ca4CaM
GoaGTP
AC5Ca
CaMKII
AC5CaGoaGTP
PP2BCaM
PDE4
AC5GoaGTP
Gabc
cAMP
PDE1CaM
PKA
PP2BCa4CaM
PP2B
Ca2CaM
CaMKIIpCa4CaM
AMP
PKAcAMP2
PKAr
Cdk5
PKAc
DARPP32
PP2BCa2CaM
Ca4CaM
CaMKII
PDE4
cAMP
PDE1CaM
PKA
PP2BCa4CaM
CaMKIICa4CaM
GabcD1R
CaMKIIpCa4CaM
PP2A
AMP
75%
PDE1
AC5CaGoaGTP
CaMKIICa4CaM
GabcD1R
PKAcAMP2
PKAr
Cdk5
PKAc
DARPP32
PP2A
75%
ATP
ATP
50%
25%
CaM
GoaGDP
AC5GoaGTP
Gabc
Ca
DaD1RGabc
50%
PKAcAMP4
PP2Ac
pThr75
PP2Ap
pThr34
25%
10%
10%
no response
no response
pThr75PKc
PKAcAMP4
PP2Ac
pThr75
PP2Ap
pThr34
pThr75PKc
Figure 8. Modularity in signal transmission. Species nodes are colored according to
their saturation/depletion status (EC90/EC10) in response to percentage of input. A.
Input is Ca (100nM...10µM), Da is set at 5µM. The figure shows a large number of species
which have no response (less than 10%) to Ca input, two separate pathways are apparent.
Species inactivate at low input levels in the ’integration zone’. B. Input is Da
(20nM...5µM), Ca is set at 8µM. The graph shows input-dependent inactivation of links.
Individual effects can be studied. For instance, PDE4 (in contrast to PDE1, PDE1CaM)
shows responsivity to Da input because of a larger complex formation with cAMP, which
subtracts from the enzyme concentration.
26
Da
D1R
DaD1R
AC5
Ca
CaM
PP2B
DaD1RGabc
Gbc
GoaGTP
AC5Ca
PP2BCaM
Ca2CaM
GoaGDP
AC5GoaGTP
Gabc
PDE1
PP2BCa2CaM
Ca4CaM
CaMKII
AC5CaGoaGTP
PDE4
cAMP
PDE1CaM
PKA
PP2BCa4CaM
CaMKIICa4CaM
GabcD1R
CaMKIIpCa4CaM
AMP
PKAcAMP2
PKAr
Cdk5
PKAc
DARPP32
PP2A
75%
ATP
50%
25%
PKAcAMP4
PP2Ac
pThr75
PP2Ap
pThr34
10%
no response
pThr75PKc
Supplemental Figure 1. Graph representation of signal transmission. Input
concentration for Da is 20nM . . . 5µM, for Ca is 100nM. There is widespread
interconnectedness for low inputs.
27
Da
D1R
DaD1R
AC5
Ca
CaM
PP2B
DaD1RGabc
Gbc
GoaGTP
AC5Ca
PP2BCaM
Ca2CaM
GoaGDP
AC5GoaGTP
Gabc
PDE1
PP2BCa2CaM
Ca4CaM
CaMKII
AC5CaGoaGTP
PDE4
cAMP
PDE1CaM
PKA
PP2BCa4CaM
CaMKIICa4CaM
GabcD1R
CaMKIIpCa4CaM
AMP
PKAcAMP2
PKAr
Cdk5
PKAc
DARPP32
PP2A
75%
ATP
50%
25%
PKAcAMP4
PP2Ac
pThr75
PP2Ap
pThr34
10%
no response
pThr75PKc
Supplemental Figure 2. Graph representation of signal transmission. Input
concentration for Da is 100nM, for Ca is 100nM . . . 10µM. The Da/GPCR pathway is
clearly separated from Ca inputs.